**COMPOUNDED ANNUAL GROWTH RATE DISTRIBUTION**

Invest for the long term.

Learning objectives:

Introduce the concept of risk by looking at the return distribution of an asset and illustrate the reduction of risk over time.

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Period: Jan 1928 - 0

Duration: 0

Minimum: 0

Maximum: 0

Average: 0

Start year of the worst period: 0

Start year of the best period: 0

The graph on this page illustrates the return distribution of the stock and bond assets. Actually, we believe that presenting the returns in a histogram form showing for example how many years stocks had a 5% return, or how many years they had a -10% return is a good way to illustrate an asset behaviour and more specifically from a risk point of view. We believe in particular that distribution range [min, max] does a good job in illustrating the risk of an asset.

Then by modifying the period length, you can see how the range changes and have a better idea of the effect of time on the risk level.

Before going to the risk characteristic of an asset, we should spend a little bit more time on the return aspect. In the previous section we made some observations using graphs and variations over selected periods of time. In order to be able to better compare the performances of the different investment assets, in finance we tend to standardize the returns and the risks by using common measurements.

A very often used measurement of an asset return is the CAGR.

First of all let's make sure that CAGR is well understood. CAGR stands for Compounded Annual Growth Rate and is used to indicate an asset return rate. This parameter should not be confused with arithmetic average return that is often used where it should not. CAGR is a better indicator to represent an asset return average, and here is why. Let's assume you had an investment that went up 100% the first year and then came down 50% the next year, an arithmetic average would give you an annual average return of 25% (100%-50%)/2, but obviously this is not the case since you ended up with the same amount as the one initially invested. CAGR describes the rate at which an investment would have grown if it grew at a steady annual rate. And here is the math CAGR (n years) = (Ending Value / Beginning Value)^{1/n} -1. CAGR, Geometric Mean and Annualized Return are all the same and differ from Arithmetic Mean that is unfortunately the more intuitive but the less useful to evaluate average return year over year.

Note that CAGR is always less than arithmetic average.

The slider below the graph allows you to modify the investment horizon during which the CAGR is computed. Move the cursor of the slider using your mouse or arrows on the keyboard. By shifting the cursor to the right you increase the investment horizon.

On the right hand side you can see the covered period, the selected period length, the minimum return, maximum return, the CAGR of the selected asset, the start year of the worst period and the best period.

First of all, take note that the CAGR for stock and bond assets are about 6% and 2% respectively based on historical data going from beginning 1928 to end of 2010. It would probably be a good idea to memorize those numbers.

When you select 1-year period, switch between bond and stock and see the differences in their corresponding histogram shapes. You can immediately see that the stocks’ returns are a lot more scattered than bonds’ returns. Furthermore stocks’ returns are ranging from about -38% to 54%, whereas bonds’ returns are ranging from about -13% to 28%. No doubt than on such short time period, stocks are more risky than bonds. It is interesting to notice though that bonds’ returns can also be quite volatile.

Now increase the time horizon by shifting gradually the cursor to the right and observe how the CAGR distribution becomes more concentrated around its average.

When you select the 30-year period, compare again the bond and the stock returns’ distribution shapes. There are two important observations to make here based on the returns’ distribution ranges for stock and bonds which are [4.3%,10.0%] and [-1.4%,5.6%] respectively. First the range’s width of the stocks’ returns is now very similar to the one for bonds’ returns, it is even smaller. This means than over a long period of time we cannot claim anymore that stocks are riskier than bonds. Second the bonds’ minimum return is negative which means than there was an erosion of the purchasing power (remember that those returns are real returns meaning that they are calculated after inflation).

We created interactive graphs to encourage you to modify the parameters, make your own observations and conclusions.

Stocks asset correspond the U.S. large-cap stock (S&P Composite), bond asset is based on the long term U.S. Treasury Bonds (10-years bond) and cash asset is based on the U.S Treasury Bills (3-month bills). The raw data for Treasury bond and bill returns is obtained from the Federal Reserve database in St. Louis (FRED). The Treasury bill rate is a 3-month rate and the Treasury bond is the constant maturity 10-year bond, but the Treasury bond return includes coupon and price appreciation. It will not match the Treasury bond rate each period

Data come from the Damodaran Online website, maintained by Aswath Damodaran, Professor of Finance at the Stern School of Business at New York University.

As you may have already noticed, the historical assets’ returns shown on this graph are after inflation return. This is different from most of the graphs showing asset return variations you may have seen before. Considering returns after inflation is a lot more relevant since it results in real returns and thus it is directly related to your purchasing power. Inflation is based on the Consumer Price Index published by the U.S Department of Labor.

Furthermore returns correspond to total returns, meaning that they have been calculated assuming that stocks dividends and interest payment have been reinvested. About the dividends, it is important to know that they have contributed more to the total return of the stock than the stock’s price appreciation.

Note also that friction costs such as transaction costs and taxes have not been taken into account but this is an acceptable limitation since it will not change significantly our comparison analysis in terms of risk and returns.

Financial models often assumed that stock returns follow a Normal Distribution (or Lognormal) also called the bell shape. This graph allows you to have a better grasp of the validity of this assumption. Actually when you use a long investment horizon, let's say 30 years you can see that historically the stock returns have a good fit with the bell shape. But this is not the case anymore when you reduce the investment horizon, and if you use annual returns for example, it clearly appears that stock returns are negatively skewed, meaning that they have exceptionally large negative returns that should not happen if Normal Distribution approximation was valid.

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